'Napoleon's Theorem' printed from http://nrich.maths.org/

Triangle $ABC$ has equilateral triangles drawn on its edges. Points
$P$, $Q$ and $R$ are the centres of the equilateral triangles. You
can change triangle $ABC$ below by dragging the vertices and
observe what happens to triangle $PQR$.

There are many ways of proving this result. One way you might
like to try involves tessellation.

(1) Draw any triangle, with angles $A, B$ and $C$ say.

(2) Draw equilateral triangles $T_1, T_2$ and $T_3$ on the three
sides of $\Delta ABC$.

(3) Fit copies of the original triangle and $T_1, T_2$ and $T_3$
into a tessellation pattern so that, at each vertex of the
tessellation, the angles are $A, B$ and $C$ and three angles of
$60^o$ making an angle sum of $360^o$.

(4) Napoleon's Theorem can be proved by simple geometry using a
small part of this pattern without even assuming that this
tessellation extends indefinitely in all directions, which is
intuitively obvious but requires advanced mathematics to prove
it.

This text is usually replaced by the Flash movie.

Van Aubel's Theorem is related to
Napoleon's Theorem. Van Aubel's Theorem states that if four squares
are drawn on the edges of any quadrilateral then the lines
joining the centres of the squares on opposite edges are equal in
length and perpendicular.